#6489634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,33)(25,36)(30,39)(31,40)(32,41)(35,42)(37,44)(38,43)(45,46)(47,48);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19,23,31)(18,24,26,41)(20,35,37,30)(21,32,22,33)(25,42,43,39)(27,40,28,29)(34,45,44,48)(36,47,38,46);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,20,23,37)(18,25,26,43)(19,30,31,35)(21,36,22,38)(24,39,41,42)(27,34,28,44)(29,45,40,48)(32,46,33,47);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(18,26)(19,31)(25,43)(27,28)(30,35)(32,33)(34,44)(46,47);(2,10)(7,16)(8,9)(14,15)(17,21,23,22)(18,27,26,28)(19,32,31,33)(20,36,37,38)(24,40,41,29)(25,34,43,44)(30,46,35,47)(39,48,42,45)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396830);